Why Markov chains, Brownian motion and the Metropolis algorithm

• We want to study a physical system which evolves towards equilibrium, from given initial conditions.
• We start with a PDF $w(x_0,t_0)$ and we want to understand how the system evolves with time.
• We want to reach a situation where after a given number of time steps we obtain a steady state. This means that the system reaches its most likely state (equilibrium situation)
• Our PDF is normally a multidimensional object whose normalization constant is impossible to find.
• Analytical calculations from $w(x,t)$ are not possible.
• To sample directly from from $w(x,t)$ is not possible/difficult.
• The transition probability $W$ is also not known.
• How can we establish that we have reached a steady state? Sounds impossible!

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